Fourier Transform¶
Images are samples of different intensities that can be represented in the frequency domain. Frequency representations enable powerful analysis and manipulation techniques.
Lesson Objectives¶
Using sines and cosines to reconstruct a signal
The Fourier Transform and its properties
Frequency domain representation of signals and images
Three properties of convolution (commutativity, associativity, distributivity)
Fourier Transform¶
The Fourier Transform decomposes a signal (or image) into a sum of sinusoidal components at different frequencies, amplitudes, and phases. This reveals which frequencies dominate the signal.
Low frequencies correspond to smooth, slowly varying regions
High frequencies correspond to edges and fine detail
Filtering in the frequency domain (e.g., low-pass, high-pass) is equivalent to convolution in the spatial domain
Blending¶
Frequency-domain analysis informs blending strategies — e.g., blending low frequencies from one image with high frequencies from another (hybrid images).
Pyramids¶
Gaussian and Laplacian pyramids provide a multi-scale frequency decomposition useful for blending, compression, and analysis.
Cuts¶
Graph cuts and min-cut/max-flow algorithms find optimal seams for compositing images.
Features¶
Feature detection identifies distinctive points in images for matching, alignment, and recognition.
Feature Detection and Matching¶
Techniques for detecting and matching features across images:
Harris corner detector: Finds points with large intensity variation in multiple directions
SIFT (Scale-Invariant Feature Transform): Detects and describes features invariant to scale and rotation
Feature matching: Comparing descriptors between images to find correspondences